Finally, we recall the construction of Householder reflectors [1]}, [2]} from numerical linear algebra, as they will play important roles in our subsequent discussions. Given a vector \(v\in \mathbb {R}^n\) , how can we build an orthogonal matrix \(H\)  such that \(Hv= \Vert v\Vert  e_1\) ? This is exactly the problem addressed by Householder reflectors, defined here as
\(H(v) \overset{\text{def}}{=}-\operatorname{sign}(v_1) \Big (I- 2\, \frac{uu^{\hspace{-0.83328pt}\mathsf {T}}}{u^{\hspace{-0.83328pt}\mathsf {T}}u}\Big ),\) 
